After decomposing the spin chain into blocks and obtaining the renormalization group flow of the coupling constants, some questions naturally arise: what are the tripartite entanglements between the three spin blocks in the low-lying energy eigenstates? How the tripartite entanglement vary as a function of the size of the blocks? And more importantly, can one uses the tripartite entanglement as a probe of the quantum phase transitions? To answer these questions, one needs to first classify and quantify entanglements for tripartite qutrit states. Although multi-partite entanglements are notoriously difficult to quantify due to the fact that almost all polynomial entanglement measures have a degree of at least four [1]}, [2]}, one may still measure genuine tripartite entanglement by the hyperdeterminant [3]}, which is a generalized determinant for higher-dimensional matrices [4]}, [5]}. Genuine multipartite entangled states are those states with a nonzero hyperdeterminant [3]}.
